I

= iron path length

If the iron permeability is high relative to the air gap (m.rot, mr stat 77 mo), the iron reluctance terms are insignificant and the following equation can be obtained for the flux:

Ni/o

b. Force calculation—The basic force equation for an air gap is

Fgap 2/o w

This equation assumes negligible leakage and fringing and that the flux density is uniform in the air gap. The combined vector force along the center of the bearing for the two air gaps of the top magnet is

Substituting from Eq. 4 gives

If the saturation flux density, Bsat, of the iron material is used for B, Eq. 6 defines the load capacity as a function of pole area for a heteropolar magnetic bearing.

c. Linearization of the force/current characteristic—Substituting from Eq. 3 gives:

FtoP = cos a — (—— | = cos a AN2/o — = k — (7)

— ( -— ) = cos a AN2mo -2 = ki — mo\ g J g2 g kl = cos a AN2mo

Thus, the force in a given magnet is proportional to the square of the current, a result that makes the bearing more difficult to control. Additionally, a single electromagnet can only apply a force in one direction (an attractive force). For these two reasons, opposed electromagnets are used together with a bias current (or flux) in each coil.

The current relationship is to increase the force, con to decrease the force.

The rotor may also be off-center in the air gap, described by

FIGURE 11 Force versus current for radial bearing of multistage pump

Current (amps)

FIGURE 11 Force versus current for radial bearing of multistage pump g = g0 - y top air gap g = g0 + y bottom air gap Applying Eq. 7 to both top and bottom electromagnets yields

With the rotor centered (y = 0), this can be reduced to

Thus the net bearing force is proportional to the control current. Figures 11 and 12 are examples of measured force versus current for radial and axial thrust bearings respectively. These measurements were made on the magnetic bearing of the multistage pump in Table 1 (refer to Figures 1 and 2).

d. Force constant and negative stiffness—Eq. 8 can be also be linearized for small motion about the center (y << g0) by differentiating with respect to icon and y, the two quantities in Eq. 8 that can change in normal operation of the bearing. The result is

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